Algebraic setFrom Wikipedia, the free encyclopedia

Contents

1 Abelian extension 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Algebraic closure 22.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Existence of an algebraic closure and splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Separable closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Algebraic element 43.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Algebraic extension 64.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Algebraic number eld 85.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.1.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3 Algebraicity and ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.3.1 Unique factorization and class number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3.2 -functions, L-functions and class number formula . . . . . . . . . . . . . . . . . . . . . . 10

5.4 Bases for number elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4.1 Integral basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4.2 Power basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

i

ii CONTENTS

5.5 Regular representation, trace and determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.6 Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6.1 Archimedean places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6.2 Nonarchimedean or ultrametric places . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6.3 Prime ideals in OF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.7 Ramication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.7.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.7.2 Dedekind discriminant theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.8 Galois groups and Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.9 Local-global principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.9.1 Local and global elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.9.2 Hasse principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.9.3 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Algebraic variety 196.1 Introduction and denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.1.1 Ane varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.1.2 Projective varieties and quasi-projective varieties . . . . . . . . . . . . . . . . . . . . . . . 216.1.3 Abstract varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2.1 Subvariety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2.2 Ane variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2.3 Projective variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.3 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4 Isomorphism of algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.5 Discussion and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.6 Algebraic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 All one polynomial 277.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Almost linear hash function 29

CONTENTS iii

8.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Characteristic (algebra) 319.1 Other equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2 Case of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.3 Case of elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10 Countable set 3410.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11 Degenerate bilinear form 4211.1 Non-degenerate forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.2 Using the determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.3 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.5 Innite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.6 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12 Degree of a eld extension 4412.1 Denition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.2 The multiplicativity formula for degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

12.2.1 Proof of the multiplicativity formula in the nite case . . . . . . . . . . . . . . . . . . . . 4512.2.2 Proof of the formula in the innite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

13 Dual basis in a eld extension 47

14 Field extension 4814.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

iv CONTENTS

14.5 Algebraic and transcendental elements and extensions . . . . . . . . . . . . . . . . . . . . . . . . 4914.6 Normal, separable and Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.8 Extension of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15 Field trace 5215.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.3 Properties of the trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.4 Finite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

15.4.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.5 Trace form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16 Finite eld 5616.1 Denitions, rst examples, and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.3 Explicit construction of nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

16.3.1 Non-prime elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.3.2 Field with four elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.3.3 GF(p2) for an odd prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.3.4 GF(8) and GF(27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.3.5 GF(16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

16.4 Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.4.1 Discrete logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.4.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

16.5 Frobenius automorphism and Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.6 Polynomial factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

16.6.1 Irreducible polynomials of a given degree . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.6.2 Number of monic irreducible polynomials of a given degree over a nite eld . . . . . . . . 63

16.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

16.8.1 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.8.2 Wedderburns little theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

CONTENTS v

16.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6416.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6416.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6516.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

17 Galois extension 6617.1 Characterization of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

18 Ideal (ring theory) 6818.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6918.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6918.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.6 Ideal generated by a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

18.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.7 Types of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.8 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.9 Ideal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.10Ideals and congruence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

19 Linear map 7419.1 Denition and rst consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7519.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7519.4 Examples of linear transformation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7719.5 Forming new linear maps from given ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.6 Endomorphisms and automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.7 Kernel, image and the ranknullity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.8 Cokernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

19.8.1 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.9 Algebraic classications of linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.10Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.11Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.12Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vi CONTENTS

19.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

20 Minimal polynomial (eld theory) 8320.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

20.1.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8420.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

21 Monic polynomial 8521.1 Univariate polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

21.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

21.2 Multivariate polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8721.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

22 Normal extension 8822.1 Equivalent properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8822.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.3 Normal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

23 Perfect eld 9023.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9023.2 Field extension over a perfect eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.3 Perfect closure and perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

24 Purely inseparable extension 9324.1 Purely inseparable extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

24.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9324.2 Galois correspondence for purely inseparable extensions . . . . . . . . . . . . . . . . . . . . . . . 9424.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9424.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

25 Ring homomorphism 9625.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9625.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9725.3 The category of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

25.3.1 Endomorphisms, isomorphisms, and automorphisms . . . . . . . . . . . . . . . . . . . . . 98

CONTENTS vii

25.3.2 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

26 Root of unity 9926.1 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10026.2 Elementary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10026.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10126.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.5 Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.6 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10626.7 Cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.8 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10826.9 Cyclotomic elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10926.10Relation to integer rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10926.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10926.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11026.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11126.14Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

27 Separable extension 11227.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11227.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11327.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11327.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.5 The denition of separable non-algebraic extension elds . . . . . . . . . . . . . . . . . . . . . . 11427.6 Dierential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11627.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

28 Separable polynomial 11728.1 Older denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11728.2 Separable eld extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11728.3 Applications in Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11828.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11828.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

29 Simple extension 11929.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11929.2 Structure of simple extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

viii CONTENTS

29.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12029.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

30 Splitting eld 12130.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130.2 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130.3 Constructing splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

30.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130.3.2 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130.3.3 The eld Ki[X]/(f(X)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

30.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.4.1 The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.4.2 Cubic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.4.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

30.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12430.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12430.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

31 Tower of elds 12531.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12531.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

32 Zorns lemma 12632.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12632.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12632.3 Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.5 Equivalent forms of Zorns lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.6 Pop Culture References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 130

32.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13032.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 1

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois groupis a cyclic group, we have a cyclic extension. A Galois extension is called solvable if its Galois group is solvable, i.e.if it is constructed from a series of abelian groups corresponding to intermediate extensions.Every nite extension of a nite eld is a cyclic extension. The development of class eld theory has provided detailedinformation about abelian extensions of number elds, function elds of algebraic curves over nite elds, and localelds.There are two slightly dierent concepts of cyclotomic extensions: these can mean either extensions formed byadjoining roots of unity, or subextensions of such extensions. The cyclotomic elds are examples. Any cyclotomicextension (for either denition) is abelian.If a eld K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resultingso-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n,since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th rootsof elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-directproduct. The Kummer theory gives a complete description of the abelian extension case, and the KroneckerWebertheorem tells us that if K is the eld of rational numbers, an extension is abelian if and only if it is a subeld of a eldobtained by adjoining a root of unity.There is an important analogy with the fundamental group in topology, which classies all covering spaces of a space:abelian covers are classied by its abelianisation which relates directly to the rst homology group.

1.1 References Kuz'min, L.V. (2001), cyclotomic extension, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

1

Chapter 2

Algebraic closure

For other uses, see Closure (disambiguation).

In mathematics, particularly abstract algebra, an algebraic closure of a eld K is an algebraic extension of K that isalgebraically closed. It is one of many closures in mathematics.Using Zorns lemma, it can be shown that every eld has an algebraic closure,[1][2][3] and that the algebraic closureof a eld K is unique up to an isomorphism that xes every member of K. Because of this essential uniqueness, weoften speak of the algebraic closure of K, rather than an algebraic closure of K.The algebraic closure of a eld K can be thought of as the largest algebraic extension of K. To see this, note that if Lis any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is containedwithin the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed eld containingK, because ifM is any algebraically closed eld containing K, then the elements ofM that are algebraic over K forman algebraic closure of K.The algebraic closure of a eld K has the same cardinality as K if K is innite, and is countably innite if K is nite.[3]

2.1 Examples The fundamental theorem of algebra states that the algebraic closure of the eld of real numbers is the eld ofcomplex numbers.

The algebraic closure of the eld of rational numbers is the eld of algebraic numbers.

There are many countable algebraically closed elds within the complex numbers, and strictly containing theeld of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers,e.g. the algebraic closure of Q().

For a nite eld of prime power order q, the algebraic closure is a countably innite eld that contains a copyof the eld of order qn for each positive integer n (and is in fact the union of these copies).[4]

2.2 Existence of an algebraic closure and splitting eldsLet S = ffj 2 g be the set of all monic irreducible polynomials in K[x]. For each f 2 S , introduce newvariables u;1; : : : ; u;d where d = degree(f) . Let R be the polynomial ring over K generated by u;i for all 2 and all i degree(f) . Write

f dYi=1

(x u;i) =d1Xj=0

r;j xj 2 R[x]

2

2.3. SEPARABLE CLOSURE 3

with r;j 2 R . Let I be the ideal in R generated by the r;j . By Zorns lemma, there exists a maximal idealM in Rthat contains I. Now R/M is an algebraic closure of K; every f splits as the product of the x (u;i +M) .The same proof also shows that for any subset S of K[x], there exists a splitting eld of S over K.

2.3 Separable closureAn algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separableextensions ofK withinKalg. This subextension is called a separable closure ofK. Since a separable extension of a sep-arable extension is again separable, there are no nite separable extensions of Ksep, of degree > 1. Saying this anotherway, K is contained in a separably-closed algebraic extension eld. It is essentially unique (up to isomorphism).[5]

The separable closure is the full algebraic closure if and only if K is a perfect eld. For example, if K is a eld ofcharacteristic p and if X is transcendental over K,K(X)( p

pX) K(X) is a non-separable algebraic eld extension.

In general, the absolute Galois group of K is the Galois group of Ksep over K.[6]

2.4 See also Algebraically closed eld Algebraic extension Puiseux expansion

2.5 References[1] McCarthy (1991) p.21

[2] M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp.11-12.

[3] Kaplansky (1972) pp.74-76

[4] Brawley, Joel V.; Schnibben, George E. (1989), 2.2 The Algebraic Closure of a Finite Field, Innite Algebraic Extensionsof Finite Fields, Contemporary Mathematics 95, American Mathematical Society, pp. 2223, ISBN 978-0-8218-5428-0,Zbl 0674.12009.

[5] McCarthy (1991) p.22

[6] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.

Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. ISBN 0-226-42451-0. Zbl 1001.16500.

McCarthy, Paul J. (1991). Algebraic extensions of elds (Corrected reprint of the 2nd ed.). New York: DoverPublications. Zbl 0768.12001.

Chapter 3

Algebraic element

In mathematics, if L is a eld extension of K, then an element a of L is called an algebraic element over K, or justalgebraic over K, if there exists some non-zero polynomial g(x) with coecients in K such that g(a)=0. Elements ofL which are not algebraic over K are called transcendental over K.These notions generalize the algebraic numbers and the transcendental numbers (where the eld extension is C/Q, Cbeing the eld of complex numbers and Q being the eld of rational numbers).

3.1 Examples The square root of 2 is algebraic over Q, since it is the root of the polynomial g(x) = x2 - 2 whose coecientsare rational.

Pi is transcendental over Q but algebraic over the eld of real numbers R: it is the root of g(x) = x - , whosecoecients (1 and -) are both real, but not of any polynomial with only rational coecients. (The denitionof the term transcendental number uses C/Q, not C/R.)

3.2 PropertiesThe following conditions are equivalent for an element a of L:

a is algebraic over K the eld extension K(a)/K has nite degree, i.e. the dimension of K(a) as a K-vector space is nite. (HereK(a) denotes the smallest subeld of L containing K and a)

K[a] = K(a), where K[a] is the set of all elements of L that can be written in the form g(a) with a polynomialg whose coecients lie in K.

This characterization can be used to show that the sum, dierence, product and quotient of algebraic elements overK are again algebraic over K. The set of all elements of L which are algebraic over K is a eld that sits in between Land K.If a is algebraic over K, then there are many non-zero polynomials g(x) with coecients in K such that g(a) = 0.However there is a single one with smallest degree and with leading coecient 1. This is the minimal polynomial ofa and it encodes many important properties of a.Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed.The eld of complex numbers is an example.

3.3 See also Algebraic independence

4

3.4. REFERENCES 5

3.4 References Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

Chapter 4

Algebraic extension

In abstract algebra, a eld extension L/K is called algebraic if every element of L is algebraic over K, i.e. if everyelement of L is a root of some non-zero polynomial with coecients in K. Field extensions that are not algebraic, i.e.which contain transcendental elements, are called transcendental.For example, the eld extensionR/Q, that is the eld of real numbers as an extension of the eld of rational numbers,is transcendental, while the eld extensionsC/R andQ(2)/Q are algebraic, whereC is the eld of complex numbers.All transcendental extensions are of innite degree. This in turn implies that all nite extensions are algebraic.[1] Theconverse is not true however: there are innite extensions which are algebraic. For instance, the eld of all algebraicnumbers is an innite algebraic extension of the rational numbers.If a is algebraic over K, then K[a], the set of all polynomials in a with coecients in K, is not only a ring but a eld:an algebraic extension of K which has nite degree over K. The converse is true as well, if K[a] is a eld, then a isalgebraic over K. In the special case where K =Q is the eld of rational numbers, Q[a] is an example of an algebraicnumber eld.A eld with no nontrivial algebraic extensions is called algebraically closed. An example is the eld of complexnumbers. Every eld has an algebraic extension which is algebraically closed (called its algebraic closure), but provingthis in general requires some form of the axiom of choice.An extension L/K is algebraic if and only if every sub K-algebra of L is a eld.

4.1 PropertiesThe class of algebraic extensions forms a distinguished class of eld extensions, that is, the following three propertieshold:[2]

1. If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.

2. If E and F are algebraic extensions of K in a common overeld C, then the compositum EF is an algebraicextension of K.

3. If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K.

These nitary results can be generalized using transnite induction:

1. The union of any chain of algebraic extensions over a base eld is itself an algebraic extension over the samebase eld.

This fact, together with Zorns lemma (applied to an appropriately chosen poset), establishes the existence of algebraicclosures.

6

4.2. GENERALIZATIONS 7

4.2 GeneralizationsMain article: Substructure

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding ofM into N is calledan algebraic extension if for every x in N there is a formula p with parameters inM, such that p(x) is true and the set

ny 2 N

p(y)ois nite. It turns out that applying this denition to the theory of elds gives the usual denition of algebraic extension.The Galois group of N overM can again be dened as the group of automorphisms, and it turns out that most of thetheory of Galois groups can be developed for the general case.

4.3 See also Integral element Lroths theorem Galois extension Separable extension Normal extension

4.4 Notes[1] See also Hazewinkel et al. (2004), p. 3.

[2] Lang (2002) p.228

4.5 References Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhalovna; Kirichenko, Vladimir V. (2004),Algebras, rings and modules 1, Springer, ISBN 1-4020-2690-0

Lang, Serge (1993), V.1:Algebraic Extensions, Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub.Co., pp. 223, ISBN 978-0-201-55540-0, Zbl 0848.13001

McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of elds, New York:Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001

Roman, Steven (1995), Field Theory, GTM 158, Springer-Verlag, ISBN 9780387944081 Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 9780130878687

Chapter 5

Algebraic number eld

In mathematics, an algebraic number eld (or simply number eld) F is a nite degree (and hence algebraic) eldextension of the eld of rational numbersQ. Thus F is a eld that containsQ and has nite dimension when consideredas a vector space over Q.The study of algebraic number elds, and, more generally, of algebraic extensions of the eld of rational numbers, isthe central topic of algebraic number theory.

5.1 Denition

5.1.1 Prerequisites

Main articles: Field and Vector space

The notion of algebraic number eld relies on the concept of a eld. A eld consists of a set of elements togetherwith two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent exampleof a eld is the eld of rational numbers, commonly denoted Q, together with its usual operations of addition etc.Another notion needed to dene algebraic number elds is vector spaces. To the extent needed here, vector spacescan be thought of as consisting of sequences (or tuples)

(x1, x2, ...)

whose entries are elements of a xed eld, such as the eld Q. Any two such sequences can be added by adding theentries one per one. Furthermore, any sequence can be multiplied by a single element c of the xed eld. These twooperations known as vector addition and scalar multiplication satisfy a number of properties that serve to dene vectorspaces abstractly. Vector spaces are allowed to be innite-dimensional, that is to say that the sequences constitutingthe vector spaces are of innite length. If, however, the vector space consists of nite sequences

(x1, x2, ..., xn),

the vector space is said to be of nite dimension, n.

5.1.2 Denition

An algebraic number eld (or simply number eld) is a nite degree eld extension of the eld of rational numbers.Here its dimension as a vector space over Q is simply called its degree.

8

5.2. EXAMPLES 9

5.2 Examples The smallest and most basic number eld is the eldQ of rational numbers. Many properties of general numberelds, such as unique factorization, are modelled after the properties of Q.

The Gaussian rationals, denoted Q(i) (read as "Q adjoined i"), form the rst nontrivial example of a numbereld. Its elements are expressions of the form

a+bi

where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added,subtracted, andmultiplied according to the usual rules of arithmetic and then simplied using the identity

i2 = 1.

Explicitly,

(a + bi) + (c + di) = (a + c) + (b + d)i,(a + bi) (c + di) = (ac bd) + (ad + bc)i.

Non-zero Gaussian rational numbers are invertible, which can be seen from the identity

(a+ bi)

a

a2 + b2 ba2 + b2

i

=

(a+ bi)(a bi)a2 + b2

= 1:

It follows that the Gaussian rationals form a number eld which is two-dimensional as a vector spaceover Q.

More generally, for any square-free integer d, the quadratic eld

Q(d)

is a number eld obtained by adjoining the square root of d to the eld of rational numbers. Arithmeticoperations in this eld are dened in analogy with the case of gaussian rational numbers, d = 1.

Cyclotomic eld

Q(n), n = exp (2i / n)

is a number eld obtained from Q by adjoining a primitive nth root of unity n. This eld containsall complex nth roots of unity and its dimension over Q is equal to (n), where is the Euler totientfunction.

The real numbers,R, and the complex numbers,C, are elds which have innite dimension asQ-vector spaces,hence, they are not number elds. This follows from the uncountability of R and C as sets, whereas everynumber eld is necessarily countable.

The set Q2 of ordered pairs of rational numbers, with the entrywise addition and multiplication is a two-dimensional commutative algebra over Q. However, it is not a eld, since it has zero divisors:

(1, 0) (0, 1) = (1 0, 0 1) = (0, 0).

10 CHAPTER 5. ALGEBRAIC NUMBER FIELD

5.3 Algebraicity and ring of integersGenerally, in abstract algebra, a eld extension F / E is algebraic if every element f of the bigger eld F is the zeroof a polynomial with coecients e0, ..., em in E:

p(f) = emfm + emfm1 + ... + e1f + e0 = 0.

It is a fact that every eld extension of nite degree is algebraic (proof: for x in F simply consider 1, x, x2, x3, ..., weget a linear dependence, i.e. a polynomial that x is a root of!) because of the nite degree. In particular this appliesto algebraic number elds, so any element f of an algebraic number eld F can be written as a zero of a polynomialwith rational coecients. Therefore, elements of F are also referred to as algebraic numbers. Given a polynomialp such that p(f) = 0, it can be arranged such that the leading coecient em is one, by dividing all coecients byit, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rationalcoecients. If, however, its coecients are actually all integers, f is called an algebraic integer. Any (usual) integerz Z is an algebraic integer, as it is the zero of the linear monic polynomial:

p(t) = t z.

It can be shown that any algebraic integer that is also a rational number must actually be an integer, whence thename algebraic integer. Again using abstract algebra, specically the notion of a nitely generated module, it canbe shown that the sum and the product of any two algebraic integers is still an algebraic integer, it follows that thealgebraic integers in F form a ring denoted OF called the ring of integers of F. It is a subring of (that is, a ringcontained in) F. A eld contains no zero divisors and this property is inherited by any subring. Therefore, the ringof integers of F is an integral domain. The eld F is the eld of fractions of the integral domain OF. This way onecan get back and forth between the algebraic number eld F and its ring of integers OF. Rings of algebraic integershave three distinctive properties: rstly, OF is an integral domain that is integrally closed in its eld of fractions F.Secondly, OF is a Noetherian ring. Finally, every nonzero prime ideal of OF is maximal or, equivalently, the Krulldimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (orDedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.

5.3.1 Unique factorization and class numberFor general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product ofprime ideals. However, unlike Z as the ring of integers of Q, the ring of integers of a proper extension of Q neednot admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. Thishappens already for quadratic integers, for example in OQ = Z[5], the uniqueness of the factorization fails:

6 = 2 3 = (1 + 5) (1 5).

Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors donot just dier by a unit in OQ. Euclidean domains are unique factorization domains; for example Z[i], the ringof Gaussian integers, and Z[], the ring of Eisenstein integers, where is a third root of unity (unequal to 1), havethis property.[1]

5.3.2 -functions, L-functions and class number formulaThe failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of theso-called ideal class group. This group is always nite. The ring of integers OF possesses unique factorization if andonly if it is a principal ring or, equivalently, if F has class number 1. Given a number eld, the class number is oftendicult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginaryquadratic number elds (i.e., Q(d), d 1) with prescribed class number. The class number formula relates h toother fundamental invariants of F. It involves the Dedekind zeta function F(s), a function in a complex variable s,dened by

F (s) :=Yp

1

1N(p)s

5.4. BASES FOR NUMBER FIELDS 11

(The product is over all prime ideals of OF, N(p) denotes the norm of the prime ideal or, equivalently, the (-nite) number of elements in the residue eld OF /p . The innite product converges only for Re(s) > 1, in generalanalytic continuation and the functional equation for the zeta-function are needed to dene the function for all s). TheDedekind zeta-function generalizes the Riemann zeta-function in that Q(s) = (s).The class number formula states that F(s) has a simple pole at s = 1 and at this point (its meromorphic continuationto the whole complex plane) the residue is given by

2r1 (2)r2 h Regw pjDj :

Here r1 and r2 classically denote the number of real embeddings and pairs of complex embeddings of F, respectively.Moreover, Reg is the regulator of F, w the number of roots of unity in F and D is the discriminant of F.Dirichlet L-functions L(, s) are a more rened variant of (s). Both types of functions encode the arithmetic behaviorof Q and F, respectively. For example, Dirichlets theorem asserts that in any arithmetic progression

a, a + m, a + 2m, ...

with coprime a andm, there are innitely many prime numbers. This theorem is implied by the fact that the DirichletL-function is nonzero at s = 1. Using much more advanced techniques including algebraic K-theory and Tamagawameasures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture),of values of more general L-functions.[2]

5.4 Bases for number elds

5.4.1 Integral basisAn integral basis for a number eld F of degree n is a set

B = {b1, , bn}

of n algebraic integers in F such that every element of the ring of integers OF of F can be written uniquely as aZ-linear combination of elements of B; that is, for any x in OF we have

x = m1b1 + + mnbn,

where the mi are (ordinary) integers. It is then also the case that any element of F can be written uniquely as

m1b1 + + mnbn,

where now the mi are rational numbers. The algebraic integers of F are then precisely those elements of F where themi are all integers.Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis,and it is now standard for computer algebra systems to have built-in programs to do this.

5.4.2 Power basisLet F be a number eld of degree n. Among all possible bases of F (seen as a Q-vector space), there are particularones known as power bases, that are bases of the form

Bx = {1, x, x2, ..., xn1}

for some element x F. By the primitive element theorem, there exists such an x, called a primitive element. If x canbe chosen in OF and such that Bx is a basis of OF as a free Z-module, then Bx is called a power integral basis, and theeld F is called a monogenic eld. An example of a number eld that is not monogenic was rst given by Dedekind.His example is the eld obtained by adjoining a root of the polynomial x3 x2 2x 8.[3]

12 CHAPTER 5. ALGEBRAIC NUMBER FIELD

5.5 Regular representation, trace and determinant

Using the multiplication in F, the elements of the eld F may be represented by n-by-n matrices

A = A(x)=(aij) i, j n,

by requiring

xei =

nXj=1

aijej ; aij 2 Q:

Here e1, ..., en is a xed basis for F, viewed as a Q-vector space. The rational numbers aij are uniquely determinedby x and the choice of a basis since any element of F can be uniquely represented as a linear combination of the basiselements. This way of associating a matrix to any element of the eld F is called the regular representation. Thesquare matrix A represents the eect of multiplication by x in the given basis. It follows that if the element y of F isrepresented by a matrix B, then the product xy is represented by the matrix product BA. Invariants of matrices, suchas the trace, determinant, and characteristic polynomial, depend solely on the eld element x and not on the basis. Inparticular, the trace of the matrix A(x) is called the trace of the eld element x and denoted Tr(x), and the determinantis called the norm of x and denoted N(x).By denition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a linearfunction of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(x) = Tr(x), and the norm is amultiplicative homogeneousfunction of degree n: N(xy) = N(x) N(y), N(x) = n N(x). Here is a rational number, and x, y are any two elementsof F.The trace form derives is a bilinear form dened by means of the trace, as Tr(x y). The integral trace form, an integer-valued symmetric matrix is dened as t = Tr(bb), where b1, ..., b is an integral basis for F. The discriminant of Fis dened as det(t). It is an integer, and is an invariant property of the eld F, not depending on the choice of integralbasis.The matrix associated to an element x of F can also be used to give other, equivalent descriptions of algebraic integers.An element x of F is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated tox is a monic polynomial with integer coecients. Suppose that the matrix A that represents an element x has integerentries in some basis e. By the CayleyHamilton theorem, pA(A) = 0, and it follows that pA(x) = 0, so that x is analgebraic integer. Conversely, if x is an element of F which is a root of a monic polynomial with integer coecientsthen the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrixin a suitable basis of F. Note that the property of being an algebraic integer is dened in a way that is independent ofa choice of a basis in F.

5.5.1 Example

Consider F = Q(x), where x satises x3 11x2 + x + 1 = 0. Then an integral basis is [1, x, 1/2(x2 + 1)], and thecorresponding integral trace form is

24 3 11 6111 119 65361 653 3589

35:The 3 in the upper left hand corner of this matrix is the trace of the matrix of the map dened by the rst basiselement (1) in the regular representation of F on F. This basis element induces the identity map on the 3-dimensionalvector space, F. The trace of the matrix of the identity map on a 3-dimensional vector space is 3.The determinant of this is 1304 = 23163, the eld discriminant; in comparison the root discriminant, or discriminantof the polynomial, is 5216 = 25163.

5.6. PLACES 13

5.6 PlacesMathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number.[4][5] Thissituation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all ofthe various possible embeddings of a number eld F into its various topological completions at once.A place of a number eld F is an equivalence class of absolute values on F. Essentially, an absolute value is a notion tomeasure the size of elements f of F. Two such absolute values are considered equivalent if they give rise to the samenotion of smallness (or proximity). In general, they fall into three regimes. Firstly (and mostly irrelevant), the trivialabsolute value | |0, which takes the value 1 on all non-zero f in F. The second and third classes are Archimedeanplaces and non-Archimedean (or ultrametric) places. The completion of F with respect to a place is given in bothcases by taking Cauchy sequences in F and dividing out null sequences, that is, sequences (xn)n N such that |xn|tends to zero when n tends to innity. This can be shown to be a eld again, the so-called completion of F at thegiven place.For F = Q, the following non-trivial norms occur (Ostrowskis theorem): the (usual) absolute value, which givesrise to the complete topological eld of the real numbers R. On the other hand, for any prime number p, the p-adicabsolute values is dened by

|q|p = pn, where q = pn a/b and a and b are integers not divisible by p.

In contrast to the usual absolute value, the p-adic norm gets smaller when q is multiplied by p, leading to quite dierentbehavior of Qp vis--vis R.

5.6.1 Archimedean places[6][7]

For some of the details take a look at,[8] Chapter 11 C p. 108. Note in particular the standard notation r1 and r2 forthe number of real and complex embeddings, respectively (see below).Calculating the archimedean places of F is done as follows: let x be a primitive element of F, with minimal polynomial(over Q) f. Over R, f will generally no longer be irreducible, but its irreducible (real) factors are either of degreeone or two. Since there are no repeated roots, there are no repeated factors. The roots r of factors of degree one arenecessarily real, and replacing x by r gives an embedding of F into R; the number of such embeddings is equal to thenumber of real roots of f. Restricting the standard absolute value on R to F gives an archimedean absolute value onF; such an absolute value is also referred to as a real place of F. On the other hand, the roots of factors of degree twoare pairs of conjugate complex numbers, which allows for two conjugate embeddings into C. Either one of this pairof embeddings can be used to dene an absolute value on F, which is the same for both embeddings since they areconjugate. This absolute value is called a complex place of F.If all roots of f above are real (respectively, complex) or, equivalently, any possible embedding F C is actuallyforced to be inside R (resp. C), F is called totally real (resp. totally complex).

5.6.2 Nonarchimedean or ultrametric places

To nd the nonarchimedean places, let again f and x be as above. In Qp, f splits in factors of various degrees, noneof which are repeated, and the degrees of which add up to n, the degree of f. For each of these p-adically irreduciblefactors t, we may suppose that x satises t and obtain an embedding of F into an algebraic extension of nite degreeover Q. Such a local eld behaves in many ways like a number eld, and the p-adic numbers may similarly play therole of the rationals; in particular, we can dene the norm and trace in exactly the same way, now giving functionsmapping toQp. By using this p-adic norm mapNt for the place t, we may dene an absolute value corresponding to agiven p-adically irreducible factor t of degree m by ||t = |Nt()|p1/m. Such an absolute value is called an ultrametric,non-Archimedean or p-adic place of F.For any ultrametric place v we have that |x|v 1 for any x in OF, since the minimal polynomial for x has integerfactors, and hence its p-adic factorization has factors in Zp. Consequently, the norm term (constant term) for eachfactor is a p-adic integer, and one of these is the integer used for dening the absolute value for v.

14 CHAPTER 5. ALGEBRAIC NUMBER FIELD

5.6.3 Prime ideals in OFFor an ultrametric place v, the subset of OF dened by |x|v < 1 is an ideal P of OF. This relies on the ultrametricityof v: given x and y in P, then

|x + y|v max (|x|v, |y|v) < 1.

Actually, P is even a prime ideal.Conversely, given a prime ideal P of OF, a discrete valuation can be dened by setting vP(x) = n where n is thebiggest integer such that x Pn, the n-fold power of the ideal. This valuation can be turned into an ultrametric place.Under this correspondence, (equivalence classes) of ultrametric places of F correspond to prime ideals of OF. ForF = Q, this gives back Ostrowskis theorem: any prime ideal in Z (which is necessarily by a single prime number)corresponds to an non-archimedean place and vice versa. However, for more general number elds, the situationbecomes more involved, as will be explained below.Yet another, equivalent way of describing ultrametric places is by means of localizations of OF. Given an ultrametricplace v on a number eld F, the corresponding localization is the subring T of F of all elements x such that | x |v 1.By the ultrametric property T is a ring. Moreover, it contains OF. For every element x of F, at least one of x or x1is contained in T. Actually, since F/T can be shown to be isomorphic to the integers, T is a discrete valuation ring,in particular a local ring. Actually, T is just the localization of OF at the prime ideal P. Conversely, P is the maximalideal of T.Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations ona number eld.

5.7 Ramication

Schematic depiction of ramication: the bers of almost all points in Y below consist of three points, except for two points in Ymarked with dots, where the bers consist of one and two points (marked in black), respectively. The map f is said to be ramied inthese points of Y.

Ramication, generally speaking, describes a geometric phenomenon that can occur with nite-to-one maps (that is,maps f: X Y such that the preimages of all points y in Y consist only of nitely many points): the cardinality ofthe bers f1(y) will generally have the same number of points, but it occurs that, in special points y, this numberdrops. For example, the map

C C, z zn

has n points in each ber over t, namely the n (complex) roots of t, except in t = 0, where the ber consists of only oneelement, z = 0. One says that the map is ramied in zero. This is an example of a branched covering of Riemannsurfaces. This intuition also serves to dene ramication in algebraic number theory. Given a (necessarily nite)extension of number elds F / E, a prime ideal p of OE generates the ideal pOF of OF. This ideal may or may not bea prime ideal, but, according to the LaskerNoether theorem (see above), always is given by

5.8. GALOIS GROUPS AND GALOIS COHOMOLOGY 15

pOF = q1e1 q2e2 ... qmem

with uniquely determined prime ideals qi of OF and numbers (called ramication indices) ei. Whenever one rami-cation index is bigger than one, the prime p is said to ramify in F.The connection between this denition and the geometric situation is delivered by the map of spectra of rings SpecOFSpecOE. In fact, unramied morphisms of schemes in algebraic geometry are a direct generalization of unramiedextensions of number elds.Ramication is a purely local property, i.e., depends only on the completions around the primes p and qi. The inertiagroup measures the dierence between the local Galois groups at some place and the Galois groups of the involvednite residue elds.

5.7.1 An exampleThe following example illustrates the notions introduced above. In order to compute the ramication index of Q(x),where

f(x) = x3 x 1 = 0,

at 23, it suces to consider the eld extensionQ23(x) /Q23. Up to 529 = 232 (i.e., modulo 529) f can be factored as

f(x) = (x + 181)(x2 181x 38) = gh.

Substituting x = y + 10 in the rst factor gmodulo 529 yields y + 191, so the valuation | y |g for y given by g is | 191|23 = 1. On the other hand the same substitution in h yields y2 161y 161 modulo 529. Since 161 = 7 23,

|y|h = 16123 = 1 / 23.

Since possible values for the absolute value of the place dened by the factor h are not conned to integer powers of23, but instead are integer powers of the square root of 23, the ramication index of the eld extension at 23 is two.The valuations of any element of F can be computed in this way using resultants. If, for example y = x2 x 1, usingthe resultant to eliminate x between this relationship and f = x3 x 1 = 0 gives y3 5y2 + 4y 1 = 0. If insteadwe eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y,and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g andh (which are both 1 in this instance.)

5.7.2 Dedekind discriminant theoremMuch of the signicance of the discriminant lies in the fact that ramied ultrametric places are all places obtainedfrom factorizations inQpwhere p divides the discriminant. This is even true of the polynomial discriminant; howeverthe converse is also true, that if a prime p divides the discriminant, then there is a p-place which ramies. For thisconverse the eld discriminant is needed. This is the Dedekind discriminant theorem. In the example above, thediscriminant of the number eld Q(x) with x3 x 1 = 0 is 23, and as we have seen the 23-adic place ramies.The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramied place comes fromthe absolute value on the complex embedding of F.

5.8 Galois groups and Galois cohomologyGenerally in abstract algebra, eld extensionsF /E can be studied by examining theGalois groupGal(F /E), consistingof eld automorphisms of F leaving E elementwise xed. As an example, the Galois group Gal (Q(n) / Q) of thecyclotomic eld extension of degree n (see above) is given by (Z/nZ), the group of invertible elements in Z/nZ. Thisis the rst stepstone into Iwasawa theory.In order to include all possible extensions having certain properties, the Galois group concept is commonly appliedto the (innite) eld extension F / F of the algebraic closure, leading to the absolute Galois group G := Gal(F / F)

16 CHAPTER 5. ALGEBRAIC NUMBER FIELD

or just Gal(F), and to the extension F / Q. The fundamental theorem of Galois theory links elds in between F andits algebraic closure and closed subgroups of Gal (F). For example, the abelianization (the biggest abelian quotient)Gab of G corresponds to a eld referred to as the maximal abelian extension Fab (called so since any further extensionis not abelian, i.e., does not have an abelian Galois group). By the KroneckerWeber theorem, the maximal abelianextension of Q is the extension generated by all roots of unity. For more general number elds, class eld theory,specically the Artin reciprocity law gives an answer by describing Gab in terms of the idele class group. Also notableis the Hilbert class eld, the maximal abelian unramied eld extension of F. It can be shown to be nite over F, itsGalois group over F is isomorphic to the class group of F, in particular its degree equals the class number h of F (seeabove).In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is thenalso referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal(F), alsoknown as Galois cohomology, which in the rst place measures the failure of exactness of taking Gal(F)-invariants,but oers deeper insights (and questions) as well. For example, the Galois group G of a eld extension L / F actson L, the nonzero elements of L. This Galois module plays a signicant role in many arithmetic dualities, such asPoitou-Tate duality. The Brauer group of F, originally conceived to classify division algebras over F, can be recast asa cohomology group, namely H2(Gal (F), F).

5.9 Local-global principle

Generally speaking, the term local to global refers to the idea that a global problem is rst done at a local level,which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be puttogether to get back to some global statement. For example, the notion of sheaves reies that idea in topology andgeometry.

5.9.1 Local and global elds

Number elds share a great deal of similarity with another class of elds much used in algebraic geometry known asfunction elds of algebraic curves over nite elds. An example is Fp(T). They are similar in many respects, for ex-ample in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient elds of whichis the function eld in question) of curves. Therefore, both types of eld are called global elds. In accordance withthe philosophy laid out above, they can be studied at a local level rst, that is to say, by looking at the correspondinglocal elds. For number elds F, the local elds are the completions of F at all places, including the archimedeanones (see local analysis). For function elds, the local elds are completions of the local rings at all points of thecurve for function elds.Many results valid for function elds also hold, at least if reformulated properly, for number elds. However, thestudy of number elds often poses diculties and phenomena not encountered in function elds. For example, infunction elds, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function eldsoften serves as a source of intuition what should be expected in the number eld case.

5.9.2 Hasse principle

A prototypical question, posed at a global level, is whether some polynomial equation has a solution in F. If thisis the case, this solution is also a solution in all completions. The local-global principle or Hasse principle assertsthat for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solutioncan be done on all the completions of F, which is often easier, since analytic methods (classical analytic tools suchas intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can beused. This implication does not hold, however, for more general types of equations. However, the idea of passingfrom local data to global ones proves fruitful in class eld theory, for example, where local class eld theory is usedto obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completionsF can be explicitly determined, whereas the Galois groups of global elds, even of Q are far less understood.

5.10. SEE ALSO 17

5.9.3 Adeles and idelesIn order to assemble local data pertaining to all local elds attached to F, the adele ring is set up. A multiplicativevariant is referred to as ideles.

5.10 See also Dirichlets unit theorem, S-unit Kummer extension Minkowskis theorem, Geometry of numbers Chebotarevs density theorem Ray class group Decomposition group Genus eld

5.11 Notes[1] Ireland, Kenneth; Rosen, Michael (1998), A Classical Introduction to Modern Number Theory, Berlin, New York: Springer-

Verlag, ISBN 978-0-387-97329-6, Ch. 1.4

[2] Bloch, Spencer; Kato, Kazuya (1990), "L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol.I, Progr. Math. 86, Boston, MA: Birkhuser Boston, pp. 333400, MR 1086888

[3] Narkiewicz 2004, 2.2.6

[4] Kleiner, Israel (1999), Field theory: from equations to axiomatization. I, The American Mathematical Monthly 106 (7):677684, doi:10.2307/2589500, MR 1720431, To Dedekind, then, elds were subsets of the complex numbers.

[5] Mac Lane, Saunders (1981), Mathematical models: a sketch for the philosophy of mathematics, The American Mathe-matical Monthly 88 (7): 462472, doi:10.2307/2321751, MR 628015, Empiricism sprang from the 19th-century view ofmathematics as almost coterminal with theoretical physics.

[6] Cohn

[7] Conrad

[8] Cohn

5.12 References Cohn, Harvey (1988), A Classical Invitation to Algebraic Numbers and Class Fields, Universitext, New York:Springer-Verlag

Conrad, Keith http://www.math.uconn.edu/~{}kconrad/blurbs/gradnumthy/unittheorem.pdf Janusz, Gerald J. (1996), Algebraic Number Fields (2nd ed.), Providence, R.I.: American Mathematical Soci-ety, ISBN 978-0-8218-0429-2

Helmut Hasse, Number Theory, Springer Classics in Mathematics Series (2002) Serge Lang, Algebraic Number Theory, second edition, Springer, 2000 Richard A. Mollin, Algebraic Number Theory, CRC, 1999 Ram Murty, Problems in Algebraic Number Theory, Second Edition, Springer, 2005

18 CHAPTER 5. ALGEBRAIC NUMBER FIELD

Narkiewicz, Wadysaw (2004), Elementary and analytic theory of algebraic numbers, Springer Monographsin Mathematics (3 ed.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267

Neukirch, Jrgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322,Berlin, New York: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021

Neukirch, Jrgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehrender MathematischenWissenschaften 323, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66671-4, MR1737196, Zbl 1136.11001

Andr Weil, Basic Number Theory, third edition, Springer, 1995

Chapter 6

Algebraic variety

This article is about algebraic varieties. For the term variety of algebras, and an explanation of the dierencebetween a variety of algebras and an algebraic variety, see variety (universal algebra).In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic

The twisted cubic is a projective algebraic variety.

19

20 CHAPTER 6. ALGEBRAIC VARIETY

geometry. Classically, an algebraic variety was dened to be the set of solutions of a system of polynomial equations,over the real or complex numbers. Modern denitions of an algebraic variety generalize this notion in several dierentways, while attempting to preserve the geometric intuition behind the original denition.[1]:58

Conventions regarding the denition of an algebraic variety dier slightly. For example, some authors require thatan "algebraic variety" is, by denition, irreducible (which means that it is not the union of two smaller sets that areclosed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraicvarieties are called algebraic sets.The notion of variety is similar to that of manifold, the dierence being that a variety may have singular points, whilea manifold will not. In many languages, both varieties and manifolds are named by the same word.Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry byshowing that a monic polynomial (an algebraic object) in one variable with complex coecients is determined by theset of its roots (a geometric object) in the complex plane. Generalizing this result, Hilberts Nullstellensatz providesa fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz andrelated results, mathematicians have established a strong correspondence between questions on algebraic sets andquestions of ring theory. This correspondence is the specicity of algebraic geometry among the other subareas ofgeometry.

6.1 Introduction and denitions

An ane variety over an algebraically closed eld is conceptually the easiest type of variety to dene, which will bedone in this section. Next, one can dene projective and quasi-projective varieties in a similar way. The most generaldenition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that onecan construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety inthe 1950s.

6.1.1 Ane varieties

Main article: Ane variety

Let k be an algebraically closed eld and let An be an ane n-space over k. The polynomials f in the ring k[x1, ...,xn] can be viewed as k-valued functions on An by evaluating f at the points in An, i.e. by choosing values in A foreach xi. For each set S of polynomials in k[x1, ..., xn], dene the zero-locus Z(S) to be the set of points in An onwhich the functions in S simultaneously vanish, that is to say

Z(S) = fx 2 An j f(x) = 0 all for f 2 Sg :

A subset V of An is called an ane algebraic set if V = Z(S) for some S.[1]:2 A nonempty ane algebraic set Vis called irreducible if it cannot be written as the union of two proper algebraic subsets.[1]:3 An irreducible anealgebraic set is also called an ane variety.[1]:3 (Many authors use the phrase ane variety to refer to any anealgebraic set, irreducible or not[note 1])Ane varieties can be given a natural topology by declaring the closed sets to be precisely the ane algebraic sets.This topology is called the Zariski topology.[1]:2

Given a subset V of An, we dene I(V) to be the ideal of all polynomial functions vanishing on V :

I(V ) = ff 2 k[x1; ; xn] j f(x) = 0 all for x 2 V g :

For any ane algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring bythis ideal.[1]:4

6.2. EXAMPLES 21

6.1.2 Projective varieties and quasi-projective varieties

Main articles: Projective variety and Quasi-projective variety

Let k be an algebraically closed eld and let Pn be the projective n-space over k. Let f in k[x0, ..., xn] be ahomogeneous polynomial of degree d. It is not well-dened to evaluate f on points in Pn in homogeneous coor-dinates. However, because f is homogeneous, f (x0, ..., xn) = d f (x0, ..., xn), it does make sense to ask whetherf vanishes at a point [x0 : ... : xn]. For each set S of homogeneous polynomials, dene the zero-locus of S to be theset of points in Pn on which the functions in S vanish:

Z(S) = fx 2 Pn j f(x) = 0 all for f 2 Sg:

A subset V of Pn is called a projective algebraic set if V = Z(S) for some S.[1]:9 An irreducible projective algebraicset is called a projective variety.[1]:10

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.Given a subset V of Pn, let I(V) be the ideal generated by all homogeneous polynomials vanishing on V. For anyprojective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.[1]:10

A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every ane variety is quasi-projective.[2] Notice also that the complement of an algebraic set in an ane variety is a quasi-projective variety; inthe context of ane varieties, such a quasi-projective variety is usually not called a variety but a constructible set.

6.1.3 Abstract varieties

In classical algebraic geometry, all varieties were by denition quasiprojective varieties, meaning that they were opensubvarieties of closed subvarieties of projective space. For example, in Chapter 1 of Hartshorne a variety over analgebraically closed eld is dened to be a quasi-projective variety,[1]:15 but from Chapter 2 onwards, the term variety(also called an abstract variety) refers to a more general object, which locally is a quasi-projective variety, but whenviewed as a whole is not necessarily quasi-projective; i.e. it might not have an embedding into projective space.[1]:105So classically the denition of an algebraic variety required an embedding into projective space, and this embeddingwas used to dene the topology on the variety and the regular functions on the variety. The disadvantage of sucha denition is that not all varieties come with natural embeddings into projective space. For example, under thisdenition, the product P1 P1 is not a variety until it is embedded into the projective space; this is usually done bythe Segre embedding. However, any variety that admits one embedding into projective space admits many others bycomposing the embedding with the Veronese embedding. Consequently many notions that should be intrinsic, suchas the concept of a regular function, are not obviously so.The earliest successful attempt to dene an algebraic variety abstractly, without an embedding, was made by AndrWeil. In his Foundations of Algebraic Geometry, Weil dened an abstract algebraic variety using valuations. ClaudeChevalley made a denition of a scheme, which served a similar purpose, but was more general. However, it wasAlexander Grothendieck's denition of a scheme that was both most general and found the most widespread accep-tance. In Grothendiecks language, an abstract algebraic variety is usually dened to be an integral, separated schemeof nite type over an algebraically closed eld,[note 2] although some authors drop the irreducibility or the reducednessor the separateness condition or allow the underlying eld to be not algebraically closed.[note 3] Classical algebraicvarieties are the quasiprojective integral separated nite type schemes over an algebraically closed eld.

Existence of non-quasiprojective abstract algebraic varieties

One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata.[3] Nagatas examplewas not complete (the analog of compactness), but soon afterwards he found an algebraic surface that was completeand non-projective.[4] Since then other examples have been found.

6.2 Examples

22 CHAPTER 6. ALGEBRAIC VARIETY

6.2.1 SubvarietyA subvariety is a subset of a variety that is itself a variety (with respect to the structure induced from the ambientvariety). For example, every open subset of a variety is a variety. See also closed immersion.Hilberts Nullstellensatz says that closed subvarieties of an ane or projective variety are in one-to-one correspondencewith the prime ideals or homogeneous prime ideals of the coordinate ring of the variety.

6.2.2 Ane varietyExample 1

Let k = C, and A2 be the two-dimensional ane space over C. Polynomials in the ring C[x, y] can be viewed ascomplex valued functions on A2 by evaluating at the points in A2. Let subset S of C[x, y] contain a single element f(x, y):

f(x; y) = x+ y 1:

The zero-locus of f (x, y) is the set of points inA2 on which this function vanishes: it is the set of all pairs of complexnumbers (x, y) such that y = 1 x, commonly known as a line. This is the set Z( f ):

Z(f) = f(x; 1 x) 2 C2g:

Thus the subset V = Z( f ) of A2 is an algebraic set. The set V is not empty. It is irreducible, as it cannot be writtenas the union of two proper algebraic subsets. Thus it is an ane algebraic variety.

Example 2

Let k = C, and A2 be the two-dimensional ane space over C. Polynomials in the ring C[x, y] can be viewed ascomplex valued functions on A2 by evaluating at the points in A2. Let subset S of C[x, y] contain a single elementg(x, y):

g(x; y) = x2 + y2 1:

The zero-locus of g(x, y) is the set of points in A2 on which this function vanishes, that is the set of points (x,y) suchthat x2 + y2 = 1. As g(x, y) is an absolutely irreducible polynomial, this is an algebraic variety. The set of its realpoints (that is the points for which x and y are real numbers), is known as the unit circle; this name is also often givento the whole variety.

Example 3

The following example is neither a hypersurface, nor a linear space, nor a single point. LetA3 be the three-dimensionalane space over C. The set of points (x, x2, x3) for x in C is an algebraic variety, and more precisely an algebraiccurve that is not contained in any plane.[note 4] It is the twisted cubic shown in the above gure. It may be dened bythe equations

y x2 = 0z x3 = 0

The fact that the set of the solutions of this system of equations is irreducible needs a proof. The simplest results fromthe fact that the projection (x, y, z) (x, y) is injective on the set of the solutions and that its image is an irreducibleplane curve.

6.2. EXAMPLES 23

For more dicult examples, a similar proof may always be given, but may imply a dicult computation: rst aGrbner basis computation to compute the dimension, followed by a random linear change of variables (not alwaysneeded); then a Grbner basis computation for another monomial ordering to compute the projection and to provethat it is injective, and nally a polynomial factorization to prove the irreducibility of the image.

6.2.3 Projective varietyA projective variety is a closed subvariety of a projective space. That is, it is the zero locus of a set of homogeneouspolynomials that generate a prime ideal.

Example 1

The ane plane curve y2 = x3 - x. The corresponding projective curve is called an elliptic curve.

A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. Theprojective lineP1 is an example of a projective curve, since it appears as the zero locus of one homogeneous coordinatein the projective plane. For another example, rst consider the ane cubic curve:

y2 = x3 x

24 CHAPTER 6. ALGEBRAIC VARIETY

in the 2-dimensional ane space (over a eld of characteristic not two). It has the associated cubic homogeneouspolynomial equation:

y2z = x3 xz2

which denes a curve in P2 called an elliptic curve. The curve has genus one (genus formula); in particular, it is notisomorphic to the projective line P1, which has genus zero. Using genus to distinguish curves is very basic: in fact,the genus is the rst invariant one uses to classify curves (see also the construction of moduli of algebraic curves).

Example 2

LetV be a nite-dimensional vector space. The Grassmannian varietyGn(V) is the set of all n-dimensional subspacesof V. It is a projective variety: it is embedded into a projective space via the Plcker embedding:

Gn(V ) ,! P(^nV ); hb1; : : : ; bni 7! [b1 ^ ^ bn]

where bi are any set of linearly independent vectors in V, ^nV is the n-th exterior power of V and the bracket [w]means the line spanned by the nonzero vector w.TheGrassmannian variety comeswith a natural vector bundle (or locally free sheaf to be precise) called the tautologicalbundle, which is important in the study of characteristic classes such as Chern classes.

6.3 Basic results An ane algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and onlyif its coordinate ring is an integral domain.[5]:52[1]:4

Every nonempty ane algebraic set may be written uniquely as a nite union of algebraic varieties (where noneof the varieties in the decomposition is a subvariety of any other).[1]:5

The dimension of a variety may be dened in various equivalent ways. See Dimension of an algebraic varietyfor details.

6.4 Isomorphism of algebraic varietiesSee also: morphism of varieties

Let V1, V2 be algebraic varieties. We say V1 and V2 are isomorphic, and write V1 V2, if there are regular maps : V1 V2 and : V2 V1 such that the compositions and are the identity maps on V1 and V2respectively.

6.5 Discussion and generalizationsThe basic denitions and facts above enable one to do classical algebraic geometry. To be able to do more forexample, to deal with varieties over elds that are not al